Pii: s0925-2312(01)00540-9

Neurocomputing 38}40 (2001) 93}98 A bursting mechanism of chattering neurons based on Ca>-dependent cationic currents Toshio Aoyagi *, Nobuhiko Terada
, Youngnam Kang, Takeshi Kaneko, Tomoki Fukai Department of Applied Analysis and Complex Dynamical Systems,Graduate School of Informatics, Kyoto University, Kyoto 606-8501 Japan
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501 Japan Department of Physiology, Graduate School of Medicine, Kyoto University Kyoto, 606-8501 Japan Department of Morphological Brain Science, Graduate School of Medicine, Kyoto University, Kyoto 606-8501 Japan Department of Information-Communication Engineering, Tamagawa University, Tamagawagakuen 6-1-1, Machida, Tokyo 194-8610 Japan CREST, JST (Japan Science and Technology), Japan We propose an ionic conductance model that simulates fast rhythmic bursts in a single compartment neuron. In this modeling, the essential point is that Ca>-dependent cationic current,whose reversal potential is approximately !45 mV, plays a key role for generating the depolarizingafterpotential (DAP) and the doublet/triplet "ring. For the calcium dynamics, the kinetics of theextrusion and the chelation of intracellular free Ca> with Ca>-pump and bu!ering proteins istaken into account. The resulting model quite accurately predicts the experimentally observednatural pattern of the chattering behavior.  2001 Elsevier Science B.V. All rights reserved.
Keywords: Cation channel; Fast rhythmic burst; Depolarizing afterpotential (DAP); Chatteringbehavior; Gamma frequency band Synchronous activity among ensembles of neurons is a ubiquitous phenomenon observed in many regions of the brain. In particular, recent experiments suggest that * Corresponding author. Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501 Japan.
E-mail address: [email protected] (T. Aoyagi).
0925-2312/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S 09 2 5 - 2 3 1 2 ( 01 ) 005 4 0- 9 T. Aoyagi et al. / Neurocomputing 38}40 (2001) 93}98 stimulus-dependent synchrony provides an e$cient mechanism to bind a group ofneurons into a coherent ensemble [3]. It has been reported that such synchronized "ring is often associated with rhythmic discharge in the gamma frequency band (20}70Hz). Among neocortical pyramidal neurons, it has been revealed that suchrepetitive burst "rings can be generated by a certain class of neurons termed fastrhythmic bursting cellsa or chattering cella [4]. This implies that chattering cells maycontribute to the neuronal stimulus-dependent synchronization. It is believed that themechanism for generating rhythmic bursting is important for understanding themechanism underlying the gamma-band EEG oscillation and its functional role.
Under these circumstances, we focus here on an ionic mechanism underlying the generation of such rhythmic bursting in chattering cells. Some experimental resultssuggest that a persistent Na> current plays a crucial role in the generation of burst "rings [1,2,10,11]. Motivated by such results, some related theoretical models have been proposed in recent years [13]. However, we shall explore another possibility.
Here we consider the possibility that a novel subtype of Ca>-dependent cationicchannel contributes decisively to the generation of the rhythmic bursting. In fact, theexistence of such cationic currents has been demonstrated in recent experiments[5,6,8]. In this work we present a single-compartment neuronal model in which suchCa>-dependent cationic conductance is taken into account.
We consider a model of a single compartment as depicted in Fig. 1. In this model, only ionic currents essential to create bursting behavior are included. The dynamics ofthe membrane potential < are described as follows: , !I)!I! !I!  !I&.!I*!I  where C is the membrane capacitance. The injected and leak currents are denoted by  and I*, respectively. Action potentials can be generated by the interaction between the sodium current I, and the delayed recti"er potassium current I).
The calcium current I! is assumed to be activated at high voltages brought about by the action potentials. The voltage dependence of these currents is described bythe "rst-order kinetics of the gating variables, as formulated by Hodgkin andHuxley [7,9].
We include in the model two voltage-independent, calcium-activated potassium and cationic currents, I&. and I!  . The former is a well-known type of current contributing to the after hyperpolarization that results in the spike frequency adapta-tion. The related activation variable depends only on the calcium concentration. Ourcrucial assumption is the existence of the calcium-activated cationic current I!  , which contributes to the depolarizing after-potential (DAP), leading to the generationof burst "rings. The essential point is that the reversal potential of such cationiccurrents suggested by the experimental data is approximately !45 mV [10].
T. Aoyagi et al. / Neurocomputing 38}40 (2001) 93}98 Fig. 1. Left: Schematic representation of chattering neuron model. Right: The typical response to a shortcurrent pulse.
For the calcium dynamics, although a simple exponentially decaying pool model can be used, we adopt a more realistic model in which the extrusion and the chelationof intracellular free Ca> with Ca>-pump and bu!ering proteins are taken intoaccount [12]. These calcium dynamics reproduce a more natural pattern of theneuronal "rings, as provided by the experimental data. In this case, the di!erence inthe calcium sensitivity between the cationic channel and the AHP channel plays anessential role in generating of the DAP, leading to the chattering behavior undersuitable conditions.
3. Simulation results Typical responses in our model cells to long-lasting depolarizing current pulses are displayed in Fig. 2. The plots on the left illustrate that the burst frequency increaseswith the intensity of the injected current, "nally leading to tonic "ring. In the plots onthe right, it is shown that, even with a single injected current, the number of spikes perburst depends on the calcium sensitivity of the cationic channels denoted by theparameter K. In real neurons, experiments suggest that this sensitivity can be modulated by some neuro-active substances. Finally, we would like to emphasize thatthe values of the various model parameters used here are within biologically reason-able ranges.
In Fig. 3, we plot the inter-burst frequency as a function of the current intensity.
Only data concerning the doublet "rings are plotted. When the current is increased,the inter-burst frequency increases approximately linearly. As observed in experi-ments, the inter-burst frequency covers the wide range from 20to 70Hz. In contrast,except for the "rst small current range, the intra-burst interval keeps almost the T. Aoyagi et al. / Neurocomputing 38}40 (2001) 93}98 Fig. 2. Typical behaviors in response to long current pulses. (Left) The e!ect of the change of the currentintensity. (Right) The e!ect of the change of the calcium sensitivity K.
constant, less than 3 ms. Therefore, the current intensity has little e!ect on theintra-burst intervals.
We constructed a single compartment model including a novel type of calcium- dependent cationic current. The novel characteristic of this current is that its reversalpotential is about !40mV near the threshold. Consequently, our model can repro-duce the various aspects of the "ring patterns related to the chattering cells success-fully, even as a single compartment model. In this model, fast rhythmic bursts canbe generated by the interplay between the fast afterdepolarization (ADP) and thefollowing slow afterhyperpolarizaton (AHP). The ADP is produced by the calcium-dependent cationic current, while the AHP is produced by the calcium-dependentpotassium current. In response to the applied currents, the neuron generates burstsof action potentials. The inter-burst frequencies cover the gamma frequency band(20}70Hz) by changing the current intensity. In contrast, the intra-burst frequencyvaries little with the current intensity and its typical value is about 300 Hz. In this T. Aoyagi et al. / Neurocomputing 38}40 (2001) 93}98 Fig. 3. (Left) Inter-burst frequency as a function of the current intensity. (Right) Intra-burst frequency asa function of the current intensity. In both cases, only data concerning the doublet "rings are plotted.
sense, the stable doublet/triplet can be realized in our proposed mechanism. On theother hand, the number of spikes per burst can be controlled by the calcium sensitivity(Kd) of cationic channel.
Finally, let us consider the possible functional roles of the cationic channels. As we can see, even at the same intensity of the applied current, if the cationic channel'sparameters K are changed, the "ring mode can be switched, for example, from single to burst spiking. Therefore, this mode switching may play a certain role in regulatingthe information content of its spike train, because it is thought that bursts conveymore information than single spikes. As another possible function, this switching maycontribute to the switching between synchronization and desynchronization in bind-ing the group of neurons in response to external stimuli. These topics are now underinvestigation.
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T. Aoyagi is a lecturer in the Department of Applied Analysis and Complex Dynamical Systems, KyotoUniversity. He is also serving as a researcher for a project by CREST, Japan Science and Technology. Hispresent research focuses on the mechanisms of synchronization in neuronal systems and its functional rolesin information processing.

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